Difference Equations 2 20MARKS

1i) Consider the difference equation

yn+1 = ryn if yn < 0.5
yn+1 = r(1-yn) if yn > or = 0.5
where r>0 is a constant.

Write the difference equation as yn+1 = f(yn) for some function f. For what values of r does the sequence remain in [0,1] if y0 E [0,1]?

ii) Consider the case when 0 < r < 1. Show that there is a single fixed point in this case. Determine whether this fixed point is stable or unstable.

iii) Let r = 1. find the fixed points of the difference equation in this case.

iv) Now let r>1. Find the fixed points and determine if they are stable or sequence close to the fixed point in [ 0.5, 1].


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